\chapter{Models}\label{chap:models}
We adopt a totally-unsupervised approach to the problem, given its particular nature. We initially inspect the standard K-Means algorithm (Section \ref{sec:kmeans}), in order to align with the previous studies on the topic \parencite{grams, Falkena2020RevisitingTI,summer_regimes_cassou, 2011ClDy...36...19C, van_der_Wiel_2019, VANDERWIEL2019261, Bloomfield_2016}. In literature the experiments only focus on a frequentist approach to the clustering of the regimes, intended to give an high level overview of the most prominent configurations. However, we think that a daily-level hard-assignment to a particular regime represents an excessively simplistic assumption, although driven by ease of implementation.\\
Further, we see that across the studies \parencite{grams, Falkena2020RevisitingTI, 2011ClDy...36...19C, van_der_Wiel_2019} a discrepancy exists about the correct number of clusters to be considered, despite the four, well-established, theoretical configurations are often enough to accurately track the atmospheric dynamics. This drawback could also be addressed to a wrong choice of the algorithm, that necessarily has to take into account higher numbers of clusters, to track intermediate evolutions of the regimes, thus performing a more faithful assignment. For this reason, after some experiments on the framework, we decide to switch to a fully-Bayesian approach, adopting Mixture Models (Section \ref{sec:mixtures}) \parencite{article, MultipleRegimesinNorthernHemisphereHeightFieldsviaMixtureModelClustering, https://doi.org/10.1002/2016RG000538}.\\
The major complexity we introduce is justified by the following advantages:
\begin{enumerate}
    \item Major statistical interpretability of the system and its dynamics
    \item A direct choice of the number of the clusters: we can indeed either opt for the standard solution of 4 clusters, where the intermediate transitions between mixed regimes are caught by the assignments of more uniform probabilities to multiple clusters, or we can rely on ad-hoc validation measures taken directly from the probabilistic setting
    \item A real descriptive probabilistic weighting scheme to the occurrence of the regimes, not based on a simple frequency
    \item Linked to the previous point, we can infer a more truthful and calibrated quantification of the subsequent tasks in the pipeline, considering the effects deriving from multiple configurations, weighted by the related probabilities
\end{enumerate}
\input{content/chapters/sections/3_1kmeans}
\input{content/chapters/sections/3_2mixture_models}
\input{content/chapters/sections/3_3results}